Serve as models for studying phase transitions, transformations, surface phenomena
Examples:
Hard-sphere colloids
Charged-stabilized colloids
Coalescence of a colloidal liquid droplet, Aarts et al. Science (2004)
The self-assembly of colloids can be controlled by changing the shape, topology, or patchiness of the particles, by introducing attractions between particles, or by constraining them to a curved surface. From Manoharan Science (2015)
Colloidal hard spheres
True hard spheres do not exist. Colloidal hard spheres approximate the behaviour in many ways
Schematic representation of various models for hard-sphere colloids. (a) A sterically stabilized particle has surface “hairs” (not to scale). (b) A charged colloid has an electrical double layer (shaded area) that gives rise to an effective diameter (c) A microgel particle is a heavily cross-linked polymer. From Royall et al Soft Matter (2013)
Hard spheres
Idealised hard spheres are impenetrable objects.
They are characterised by the simplest of the potentials.
Phase behaviour of hard spheres
For a sphere of diameter \sigma, enormous simplification of the Boltzmann factor:
e^{-\beta U(r)} =
\begin{cases}
0 & \text{if } r < \sigma \\
1 & \text{if } r \geq \sigma
\end{cases}
This simplification arises because the potential ( U(r) ) is either infinite (forbidden region) or zero (allowed region).
For this reason, they hard spheres are entropy driven.
Constructing partition functions is then just counting counting valid configurations, linking geometry to thermodynamics.
A good example: Asakura-Oosawa depletion, that we have already seen.
Valid configurations are sphere packings:
The only relevant control parameter is the volume fraction\phi, defined as:
\phi = \frac{\pi}{6} \rho \sigma^3
where \rho=N/V is the number density.
Phase diagram of hard spheres
Since we have a single control parameter, the pahse diagram is one-dimensional.
Changing temeprature is immaterial to the free energy: it is a simple rescaling factor.
It is intuitive to guess the low density limit of hard spheres:
few spheres in a box should behave similalry to an ideal gas
It is harder to guess the high density limit
We will gradually build our understanding from low to dense packing.
Hard spheres have a single control parameter, the packing fraction.
Low packing fractions
When treating the Asakura-Osawa depletion we introduced the excluded volume. It is also also key for hard spheres.
The distance of closest approach between two identical spheres is \sigma, which corresponds to the diameter of the spheres. The excluded volume of one particle is v_{\rm ex} =4\pi\sigma^3/3
At low densities, hard spheres are isolated and so no overlaps occcur and the accesible volume for N spheres is V_{\rm accessible} = V-Nv_{\rm ex}
For phase behaviour, we need the thermodyanamic potential. At constant volume and number of particles, the parition function is a measure of the accessible volume
\mathcal{Z} = \frac{1}{N! \Lambda^{3N}} \int_{V_{\rm accessible}} d\mathbf{r}_1 \ldots d\mathbf{r}_N
where \Lambda is the thermal de Broglie wavelength \Lambda = h/\sqrt{2\pi mk_B T}.
Mass and temperature are factorising out.
Low packing fractions: entropy
We integrate over valid (non-overlapping) configurations such that |\mathbf{r}_i - \mathbf{r}_j| \geq \sigma for all i \neq j. This yields \mathcal{Z} = \dfrac{(V-Nv_{\rm ex}/2)^N}{N!\Lambda^{3N}},
with the 1/2 factor coming from the fact that we avoid double counting the excluded volume of pairs.
We immediately obtain the entropy as S = k_B \ln \mathcal{Z} = k_B \left[ N \ln(V - N v_{\rm ex}/2) - \ln N! - 3N \ln \Lambda \right]
Via Stirling approximation \ln N! = N\ln N -N we get S = k_B \left[ N \ln(V - N v_{\rm ex}/2) - (N \ln N - N) - 3N \ln \Lambda \right]
This can be rewritten as S = N k_B \left[ \ln\left( \frac{V - N v_{\rm ex}/2}{N \Lambda^3} \right) + 1 \right]
Low packing fraction: equation of state
The equation of state links the three relevant thermodynamics variables P,T and \phi
P = -\left(\dfrac{\partial F}{\partial V}\right)_{N,T} =T\left(\dfrac{\partial S}{\partial V}\right)_{N,T}= \dfrac{k_B T }{v-v_{\rm ex}/2}
This expression can be simplified (do it as an exercise) to obtain the equation of state
Inspect this term and recognise that this illustrates that the dilute limit is an ideal gas + a correction.
Virial expansion
The expression Z_{\rm comp} = 1+4\phi+O(\phi^2) is just the simplest form of the more generic virial expansion, the perturbative series
Z_{\rm comp}= 1 + B_2 \rho + B_3 \rho^2+ \dots
The B_2, B_3, \dots are known as virial coeffcients and for non-hard-sphere systems they also depend on temperature, B_2(T), B_3(T),\dots
They encode correlations (pairs, triplets and so on). In a genric setting
Z_N=\frac{1}{N!\Lambda^{3 N}} \int \cdots \int \exp \left[-\beta \sum_{i<j} U\left(r_{i j}\right)\right] d \mathbf{r}_1 \ldots d \mathbf{r}_N
can be re-written using the Mayer functionf_{i j}=e^{-\beta u\left(r_{i j}\right)}-1 and e^{-\beta \sum_{i<j} U\left(r_{i j}\right)}=\prod_{i<j}\left(1+f_{i j}\right) yielding
Z_N=\frac{1}{N!\Lambda^{3 N}} \int \cdots \int \prod_{i<j}\left(1+f_{i j}\right) d \mathbf{r}_1 \ldots d \mathbf{r}_N
Cluster expansion and second virial
One can expand the product
\prod_{i<j}\left(1+f_{i j}\right)=1+\sum_{i<j} f_{i j}+\sum_{i<j, k<l}^{\text {distinct }} f_{i j} f_{k l}+\ldots
The first term is the ideal gas, the second is clearly pair correlations (only pairwise distances).
The virial expansion can also be represented diagrammatically. For the compressibility factor ( Z_{} ), we have:
This motivates one to define
B_2(T)=-\frac{1}{2} \int f(r) d \mathbf{r}=-2 \pi \int_0^{\infty}\left[e^{-\beta U(r)}-1\right] r^2 d r
For hard spheres this results in
B_2 = \frac{2\pi}{3} \sigma^3
So Z = 1 + \dfrac{2\pi}{3} \sigma^3\rho = 1+ 4\left( \dfrac{\pi}{6} \sigma^3\rho\right)=1+4\phi as we calculated earlier.
Low density limit: key points
Important
We can simply focus on the accessible volume and ignore overlap between exclusion volumes
We can focus on the configurational entropy and extract the equation of state relating P,T,\phi
The compressibility factor in terms of the packing fraction \phi is the most condensed expression and is
Z_{\mathrm{comp}}=\frac{P V}{N k_B T}=1+4 \phi+O\left(\phi^2\right)
highlighting the first non-trivial perturbation to the ideal gass
We can see this as an instance of the more general virial and cluster expansion formalism. which can used for any fluid.
Z_{\text {comp }}=1+B_2 \rho+B_3 \rho^2+\ldots
The first nontrivial correction is the second virial coefficient
B_2(T)=-2 \pi \int_0^{\infty}\left[e^{-\beta U(r)}-1\right] r^2 d r
Dense packing
As we increase the packing fraction, the accessible (free) volume reduces rapidly and tehrmal motion is hindered.
Tight disordered (random) packing of spheres are described as jammed: link with glasses (in future lectures).
Densest packing reaches a maximum packing of around \phi_{\mathrm{rcp}} \approx 0.64: this is not unique and depends on the protocol of preparation.
Kepler’s conjecture (1611, proved only in 2017):
Densest packings are Face Centerd Cubic (FCC) or Hexagonally Cosed Packed (HCP)\phi_{\max }=\frac{\pi}{3 \sqrt{2}} \approx 0.74
Packings
Dense packing
Q: Can purely replusive spheres spheres assemble spontaneously into an FCC crystal?
YES
Entropy in AO interactions causes an effective interaction that has a minimum
Disordered packings are less efficient than ordered ones
NO
Hard sphere potential only prevents overlaps
There is no minimum in the interaction potential to favour a crystal lattice and localisation
Entropy is just disorder
The matter was hotly debated in conferences in the 1950s and got to evenly split votes multiple times
Dense packing and crystallisation
Early computer simulations ( molecular dynamics, Alder and Wainwright 1957, Monte Carlo, Wood and Jacobson (1957)) proved that that crystalisation is possible.
